Minimum Rank Problems · Symmetric minimum rank problem Determine mr(G) for any simple graph G....

Minimum RankProblems:Recent

Developments

Leslie Hogben

3 mr problems

Simple trees

Loop trees

Ditrees

Computing mrditrees

Ditree proof

Asymmetric tosymmetric

Symmetric mr

Related problems

Minimum Rank Problems:Recent Developments

Leslie Hogben

Iowa State University andAmerican Institute of Mathematics

15th ILAS Conference, CancunJune 16, 2008


Developments

Leslie Hogben

3 mr problems

Simple trees

Loop trees

Ditrees

Computing mrditrees

Ditree proof


Symmetric mr

Related problems

Three minimum rank problems

Simple trees

Loop trees

Ditrees

Computing minimum rank of ditrees

Proof of ditree theorem

Conversion of asymmetric minimum rank problem tosymmetric minimum rank problem

Symmetric minimum rank problem

Related problems


Developments

Leslie Hogben

3 mr problems

Simple trees

Loop trees

Ditrees

Computing mrditrees

Ditree proof


Symmetric mr

Related problems

General type of problemDetermine the minimum rank among a family of matricesdescribed by a given zero-nonzero pattern.

Example

Let Y1 =

∗ ∗ ∗∗ ∗ ∗∗ ∗ ∗

and Y2 =

∗ ∗ 0∗ ∗ ∗0 ∗ ∗

.

Minimum rank of matrices described by Y1 is 1.Minimum rank of matrices described by Y2 is 2.


Developments

Leslie Hogben

3 mr problems

Simple trees

Loop trees

Ditrees

Computing mrditrees

Ditree proof


Symmetric mr

Related problems

Some topics with connections to minimum rank problems:

I Jordan Canonical Form for matrices described by adigraph

I Inverse Eigenvalue Problem for matrices described by asimple graph

I spectral graph theory, singular graphs, nullity of theadjacency matrix

I biclique decompositions and the bicliquecover number(Graham-Pollack Theorem)

I eigensharp graphs

I Lovasz ϑ function

I communication complexity and minimum rank of ±1sign patterns


Developments

Leslie Hogben

3 mr problems

Simple trees

Loop trees

Ditrees

Computing mrditrees

Ditree proof


Symmetric mr

Related problems

Graphs and digraphsA square pattern can be described by a graph or digraph.

I a simple graph G = (VG ,EG ) does not allow loops ormultiple edges between the same pair of vertices

I a graph G = (VG,EG) allows loops but not multipleedges between the same pair of vertices

I a digraph D = (VD,ED) allows loops but not multiplearcs between the same ordered pair of vertices.


Developments

Leslie Hogben

3 mr problems

Simple trees

Loop trees

Ditrees

Computing mrditrees

Ditree proof


Symmetric mr

Related problems

A graph or digraph describes a family of matrices.

I S(G ) = {A ∈ Rn×n : AT = A andfor i 6= j , {i , j} ∈ EG ⇔ ai ,j 6= 0}

I S(G) = {A ∈ Rn×n : AT = A andmultiset {i , j} ∈ EG ⇔ ai ,j 6= 0}

I Q(D) = {A ∈ Rn×n : (i , j) ∈ ED ⇔ ai ,j 6= 0}


Developments

Leslie Hogben

3 mr problems

Simple trees

Loop trees

Ditrees

Computing mrditrees

Ditree proof


Symmetric mr

Related problems

Minimum rank and maximum nullity: digraphs

I mr(D) = min{rank(A) : A ∈ Q(D)}I M(D) = max{null(A) : A ∈ Q(D)}I mr(D) + M(D) = |D|

Asymmetric minimum rank problemDetermine mr(D) for any digraph D.

Early workSeries of papers by Hershkowitz and Schneider in 1993 and1994:

I Possible Jordan Canonical Forms for eigenvalue 0 ofmatrices in Q(D)

I Complete solution for the generic case.


Developments

Leslie Hogben

3 mr problems

Simple trees

Loop trees

Ditrees

Computing mrditrees

Ditree proof


Symmetric mr

Related problems

Minimum rank and maximum nullity: simple graphs

I mr(G ) = min{rank(A) : A ∈ S(G )}I M(G ) = max{null(A) : A ∈ S(G )}I mr(G ) + M(G ) = |G |

Symmetric minimum rank problemDetermine mr(G ) for any simple graph G .

Early work

I Arose from the inverse eigenvalue problem for S(G ).

I Complete solution for minimum rank of trees[Nylen, 96], [Johnson, Leal-Duarte, 99].

I Many results by more than 30 people from 2004 on.


Developments

Leslie Hogben

3 mr problems

Simple trees

Loop trees

Ditrees

Computing mrditrees

Ditree proof


Symmetric mr

Related problems

Minimum rank and maximum nullity: graphs

I mr(G) = min{rank(A) : A ∈ S(G)}I M(G) = max{null(A) : A ∈ S(G)}I mr(G) + M(G) = |G|

Symmetric minimum rank problem for graphs with loopsDetermine mr(G) for any graph G.

Early work

I Complete solution for minimum rank of loop trees[DeAlba, Hardy, Hentzel, Hogben, Wangsness, 06]


Developments

Leslie Hogben

3 mr problems

Simple trees

Loop trees

Ditrees

Computing mrditrees

Ditree proof


Symmetric mr

Related problems

Simple trees

In [Johnson, Leal-Duarte, 99]:

I ∆(T ) is the maximum of p − q such that there is a setof q vertices whose deletion leaves p paths

I the path cover number P(T ) is the minimum numberof vertex disjoint paths that cover all the vertices of T

I proved that P(T ) = M(T ) = ∆(T )


Developments

Leslie Hogben

3 mr problems

Simple trees

Loop trees

Ditrees

Computing mrditrees

Ditree proof


Symmetric mr

Related problems

Numerous algorithms compute ∆(T ) and P(T ) by usinghigh degree (≥ 3) vertices.

The following algorithms work from the outside in (startwith a pendent generalized star).

rest of the tree

v

I ∆(T ): Delete each outer high degree vertex v . Repeatas needed.

I P(T ): At each outer high degree vertex v , form a pathof v and two pendent paths to make one path, and useadditional pendent paths as needed. Repeat as needed.


Developments

Leslie Hogben

3 mr problems

Simple trees

Loop trees

Ditrees

Computing mrditrees

Ditree proof


Symmetric mr

Related problems

Example

Compute mr(T ) by computing ∆(T ) = M(T ).

1

2

3

4 5 6

8 7


Developments

Leslie Hogben

3 mr problems

Simple trees

Loop trees

Ditrees

Computing mrditrees

Ditree proof


Symmetric mr

Related problems

Example


1

2

3

4 5 6

8 7


Developments

Leslie Hogben

3 mr problems

Simple trees

Loop trees

Ditrees

Computing mrditrees

Ditree proof


Symmetric mr

Related problems

Example


1

2

3

45 6

7


Developments

Leslie Hogben

3 mr problems

Simple trees

Loop trees

Ditrees

Computing mrditrees

Ditree proof


Symmetric mr

Related problems

Example


1

2

3

5 6

7


Developments

Leslie Hogben

3 mr problems

Simple trees

Loop trees

Ditrees

Computing mrditrees

Ditree proof


Symmetric mr

Related problems

Example

Compute mr(T ) by computing ∆(T ) = M(T ):

I the six vertices {1, 2, 3, 5, 6, 7} were deleted

I there are 18 paths

I M(T ) = ∆(T ) = 18− 6 = 12

I mr(T ) = 35− 12 = 23

1

2

3

5 6

7


Developments

Leslie Hogben

3 mr problems

Simple trees

Loop trees

Ditrees

Computing mrditrees

Ditree proof


Symmetric mr

Related problems

Example

Compute mr(T ) by computing P(T ) = M(T ).


Developments

Leslie Hogben

3 mr problems

Simple trees

Loop trees

Ditrees

Computing mrditrees

Ditree proof


Symmetric mr

Related problems

Example

Compute mr(T ) by computing P(T ) = M(T ).


Developments

Leslie Hogben

3 mr problems

Simple trees

Loop trees

Ditrees

Computing mrditrees

Ditree proof


Symmetric mr

Related problems

Example

P(T ) = 12 = M(T ) and mr(T ) = 35− 12 = 23.


Developments

Leslie Hogben

3 mr problems

Simple trees

Loop trees

Ditrees

Computing mrditrees

Ditree proof


Symmetric mr

Related problems

Loop trees

A loop tree T is a graph that allows loops but not multipleedges that is connected and has no cycles of length greaterthan 1.

In [DeAlba, Hardy, Hentzel, Hogben, Wangsness, 06]:

I C0(T) = max{c0(Q)− |Q|} where c0(Q) is the numberof singular components of T− Q

I algorithm to compute C0(T) that generalizes thealgorithm for computing ∆(T ) working from theoutside in.

I proved that C0(T) = M(T)


Developments

Leslie Hogben

3 mr problems

Simple trees

Loop trees

Ditrees

Computing mrditrees

Ditree proof


Symmetric mr

Related problems

Example

Compute mr(T) by computing C0(T) = M(T).

1

2

3

4 5 6

8 7


Developments

Leslie Hogben

3 mr problems

Simple trees

Loop trees

Ditrees

Computing mrditrees

Ditree proof


Symmetric mr

Related problems

Example


1

2

3

4 5 6

8 7


Developments

Leslie Hogben

3 mr problems

Simple trees

Loop trees

Ditrees

Computing mrditrees

Ditree proof


Symmetric mr

Related problems

Example


2

3

4 5 6

8


Developments

Leslie Hogben

3 mr problems

Simple trees

Loop trees

Ditrees

Computing mrditrees

Ditree proof


Symmetric mr

Related problems

Example

M(T) = C0(T) = 12− 5 = 7 and mr(T) = 35− 7 = 28.

2

3

5 6

8


Developments

Leslie Hogben

3 mr problems

Simple trees

Loop trees

Ditrees

Computing mrditrees

Ditree proof


Symmetric mr

Related problems

The obvious generalization of path cover number fails toequal maximum nullity:

Example

The minimum number of paths needed to cover the doublepath T is 2 but M(T) = 1


Developments

Leslie Hogben

3 mr problems

Simple trees

Loop trees

Ditrees

Computing mrditrees

Ditree proof


Symmetric mr

Related problems

Ditrees

The underlying simple graph of D is the simple graphobtained by deleting loops and then replacing every arc(v ,w) or pair of arcs (v ,w), (w , v) by the edge {v ,w}.

A ditree T is a digraph whose underlying simple graph is atree.


Developments

Leslie Hogben

3 mr problems

Simple trees

Loop trees

Ditrees

Computing mrditrees

Ditree proof


Symmetric mr

Related problems

The associated loop tree of a symmetric ditree is the looptree obtained by replacing every pair of arcs (v ,w), (w , v) bythe edge {v ,w} (and arc (v , v) by edge {v , v}).

ObservationIf T is a symmetric ditree and T′ is the associated loop treethen mr(T) = mr(T′).

Note that T′ describes only symmetric matrices.


Developments

Leslie Hogben

3 mr problems

Simple trees

Loop trees

Ditrees

Computing mrditrees

Ditree proof


Symmetric mr

Related problems

Results of the AIM Minimum Rank Square in February 08[AIM Square] (Barioli, Fallat, Hall, Hershkowitz, Hogben,van der Holst, Shader):

DefinitionThe path cover number P(D) of D is the minimum numberof vertex disjoint paths whose deletion from D leaves adigraph that requires nonsingularity (or the empty set).

TheoremIf T is a ditree then P(T) = M(T).


Developments

Leslie Hogben

3 mr problems

Simple trees

Loop trees

Ditrees

Computing mrditrees

Ditree proof


Symmetric mr

Related problems

This definition of path cover number works for the doublepath:

Example

The deletion of one path from T leaves a nonsingular graphso P(T) = 1 = M(T).


Developments

Leslie Hogben

3 mr problems

Simple trees

Loop trees

Ditrees

Computing mrditrees

Ditree proof


Symmetric mr

Related problems

Example

M(T) = P(T) = 5.

How is P(T) found?


Developments

Leslie Hogben

3 mr problems

Simple trees

Loop trees

Ditrees

Computing mrditrees

Ditree proof


Symmetric mr

Related problems

Computing minimum rank of ditrees

The zero forcing number was introduced for simple graphs in[AIM 08] (18 authors, based on the AIM 2006 workshop).We adapted it to digraphs. D is a digraph with each vertexcolored either white or black.

I out color change rule: If u is a vertex of D, and exactlyone out-neighbor w of u is white, then change the colorof w to black (u forces w).

I out derived coloring: result of applying the out colorchange rule until no more changes are possible.

I out zero forcing set: Z ⊂ VD such that if Z is coloredblack, the out derived coloring is all black.

I zero forcing number: Zo(D): minimum of |Z | over allout zero forcing sets Z ⊆ VD.


Developments

Leslie Hogben

3 mr problems

Simple trees

Loop trees

Ditrees

Computing mrditrees

Ditree proof


Symmetric mr

Related problems

Theorem[AIM Square] If T is a ditree then P(T) = Zo(T).

I Zo(D) can be computed by brute force.

I ISU group (DeLoss, Grout, McKay, Smith, Tims) havea program to compute Zo(D) in Sage.

I It produced the following zero forcing set for theprevious example.


Developments

Leslie Hogben

3 mr problems

Simple trees

Loop trees

Ditrees

Computing mrditrees

Ditree proof


Symmetric mr

Related problems

Example

Verify Zo(T) ≤ 5 by finding the out derived set.


Developments

Leslie Hogben

3 mr problems

Simple trees

Loop trees

Ditrees

Computing mrditrees

Ditree proof


Symmetric mr

Related problems

Example



Developments

Leslie Hogben

3 mr problems

Simple trees

Loop trees

Ditrees

Computing mrditrees

Ditree proof


Symmetric mr

Related problems

Example



Developments

Leslie Hogben

3 mr problems

Simple trees

Loop trees

Ditrees

Computing mrditrees

Ditree proof


Symmetric mr

Related problems

Example



Developments

Leslie Hogben

3 mr problems

Simple trees

Loop trees

Ditrees

Computing mrditrees

Ditree proof


Symmetric mr

Related problems

Example



Developments

Leslie Hogben

3 mr problems

Simple trees

Loop trees

Ditrees

Computing mrditrees

Ditree proof


Symmetric mr

Related problems

Example



Developments

Leslie Hogben

3 mr problems

Simple trees

Loop trees

Ditrees

Computing mrditrees

Ditree proof


Symmetric mr

Related problems

Outline of the proof of the ditree theoremM(T) = P(T) = Zo(T) [AIM Square]

Let Z be an out zero forcing set of a digraph D. Constructthe out derived set, recording the forces.

I forcing chain: a sequence of vertices (v1, v2, . . . , vk)such that for i = 1, . . . , k − 1, vi forces vi+1.

I forcing chain digraph (of the forcing chain(v1, v2, . . . , vk)): the digraph H = (VH,EH) whereVH = {v1, v2, . . . , vk} andEH = {(v1, v2), (v2, v3), . . . , (vk−1, vk)}.

LemmaAny forcing chain digraph is a path or a cycle.


Developments

Leslie Hogben

3 mr problems

Simple trees

Loop trees

Ditrees

Computing mrditrees

Ditree proof


Symmetric mr

Related problems

Theorem (AIM Square)

For any digraph D, P(D) ≤ Zo(D).

Proof:

I Choose an out zero forcing set of order Zo(D).

I P is the set of all maximal forcing chain digraphs thatare paths.

I D− P can force itself.

I So D− P is nonsingular.

I P(D) ≤ |P| ≤ |Z | = Zo(D).


Developments

Leslie Hogben

3 mr problems

Simple trees

Loop trees

Ditrees

Computing mrditrees

Ditree proof


Symmetric mr

Related problems

We need to define path cover number without induced toobtain P(D) ≤ Zo(D).

Example


Developments

Leslie Hogben

3 mr problems

Simple trees

Loop trees

Ditrees

Computing mrditrees

Ditree proof


Symmetric mr

Related problems

The triangle number has been used to bound minimum rankof a pattern from below ([Canto, 11th ILAS], [Johnson, 12thILAS]).

I t-triangle of an m × n pattern Y : a t × t subpatternthat is permutation similar to a pattern that is uppertriangular with all diagonal entries nonzero.

I triangle number: tri(Y ) = maximum size of a trianglein Y .

I For a digraph D, tri(D) = tri(Y(D)).

I tri(D) ≤ mr(D)


Developments

Leslie Hogben

3 mr problems

Simple trees

Loop trees

Ditrees

Computing mrditrees

Ditree proof


Symmetric mr

Related problems


tri(D) + Zo(D) = |D|.Proof: Zo(D) ≤ |D| − tri(D)

I Suppose D has a t-triangle.

I The columns not in the t-triangle constitute a zeroforcing set.

I So Zo(D) ≤ |D| − tri(D).

Example0 ∗ 00 ∗ 0∗ ∗ ∗

3 1

2

3 is black, 1 forces 2 and then 3 forces 1, so Zo(D) ≤ 1.


Developments

Leslie Hogben

3 mr problems

Simple trees

Loop trees

Ditrees

Computing mrditrees

Ditree proof


Symmetric mr

Related problems

Theoremtri(D) + Zo(D) = |D|.Proof: tri(D) ≥ |D| − Zo(D)

I Let Z be an out zero forcing set.

I In Y = Y(D), delete the columns whose indices are inZ to obtain Y ′.

I Compute tri(Y ′) by elimination:I Vertex v forcing vertex w means the v ,w entry is the

only nonzero entry of row v .I Delete row v and column w and add 1 to tri(Y ′).

I Thus tri(D) ≥ tri(Y ′) = |D| − Zo(D).

Example0 ∗ 00 ∗ 0∗ ∗ ∗

→0 ∗

0 ∗∗ ∗

→ [0∗

]so tri(Y(D)) ≥ 2.


Developments

Leslie Hogben

3 mr problems

Simple trees

Loop trees

Ditrees

Computing mrditrees

Ditree proof


Symmetric mr

Related problems

Edit Distance to nonsingularityLet Y be a square pattern and let D be a digraph.

I (row) edit distance to nonsingularity, ED(Y ): theminimum number of rows that must be changed toobtain a pattern that requires nonsingularity.

I ED(D) = ED(Y(D)).

Example


Developments

Leslie Hogben

3 mr problems

Simple trees

Loop trees

Ditrees

Computing mrditrees

Ditree proof


Symmetric mr

Related problems


For any ditree T, ED(T) ≤ P(T).

Proof:

I Let P = {P1, . . . ,Pk} be a set of vertex-disjoint pathssuch that T− VP requires nonsingularity

I Let vi be the first vertex and wi the last vertex of Pi .

I Edit row wi (i.e., edit the out-neighborhood of wi ) sothat the only out-neighbor of wi is vi .

I This involves k edits and produces a digraph D.

I D requires nonsingularity, which impliesED(T) ≤ k = P(T).


Developments

Leslie Hogben

3 mr problems

Simple trees

Loop trees

Ditrees

Computing mrditrees

Ditree proof


Symmetric mr

Related problems

Example

ED(T) ≤ P(T) ≤ 2


Developments

Leslie Hogben

3 mr problems

Simple trees

Loop trees

Ditrees

Computing mrditrees

Ditree proof


Symmetric mr

Related problems


tri(D) + ED(D) = |D|.Proof is similar to tri(D) + Zo(D) = |D|.

Corollary

For a ditree, ED(T) = Zo(T) = P(T).

Proof: P(T) ≤ Zo(T) = ED(T) ≤ P(T).


Developments

Leslie Hogben

3 mr problems

Simple trees

Loop trees

Ditrees

Computing mrditrees

Ditree proof


Symmetric mr

Related problems

It remained to prove M(T) = P(T).

The AIM Square gave a complicated argument that used theloop tree result and used the triangle number to reduce theproblem for ditrees to the strong (symmetric) components.

QuestionIs P(D) ≤ M(D) for every digraph?

If so (and if proved) this would provide an alternate proof ofM(T) = P(T):P(T) ≤ M(T) ≤ Zo(T) = ED(T) ≤ P(T).


Developments

Leslie Hogben

3 mr problems

Simple trees

Loop trees

Ditrees

Computing mrditrees

Ditree proof


Symmetric mr

Related problems

Conversion of an asymmetric minimum rank problem to asymmetric minimum rank problemHere Y need not be square.


Let Y be an m × n pattern Y such that every row andcolumn of Y has a nonzero entry.

DY is the symmetric digraph having pattern

[∗ Y

Y T ∗

], and

GY is the underlying simple graph of DY . Then

mr(Y ) = mr(DY ) = mr(GY ).


Developments

Leslie Hogben

3 mr problems

Simple trees

Loop trees

Ditrees

Computing mrditrees

Ditree proof


Symmetric mr

Related problems

Symmetric minimum rank problem

Minimum rank is characterized for:

I trees [Nylen 96], [Johnson, Leal-Duarte 99]

I unicyclic graphs [Barioli, Fallat, Hogben 05]

I all small graphs (|G | ≤ 7) [ISU group]

I extreme minimum rank:mr(G ) = 0, 1, 2: [Barrett, van der Holst, Loewy 04]mr(G ) = |G | − 1, |G | − 2: [Fiedler 69],[Hogben, van der Holst 07], [Johnson, Loewy, Smith]

I many families of graphs [AIM 08]


Developments

Leslie Hogben

3 mr problems

Simple trees

Loop trees

Ditrees

Computing mrditrees

Ditree proof


Symmetric mr

Related problems

Reduction techniques for

I cut-set of order 1 [Barioli, Fallat, Hogben 04]and order 2 [van der Holst 08]

I joins [Barioli, Fallat 06]

Bounds for minimum rank/maximum nullity:

I M(G ) ≤ Z (G ) [AIM 08]

I µ(G ) ≤ M(G ) [Colin de Verdiere 93]ξ(G ) ≤ M(G ) [Barioli, Fallat, Hogben 05]


Developments

Leslie Hogben

3 mr problems

Simple trees

Loop trees

Ditrees

Computing mrditrees

Ditree proof


Symmetric mr

Related problems

Minimum rank graph catalogs

I Minimum rank of many families of graphs determinedat the 06 AIM Workshop.

I On-line catalogs of minimum rank for small graphs andfamilies developed.

I The ISU group determined the order of all graphs oforder 7.

Minimum rank of families of graphshttp://aimath.org/pastworkshops/catalog2.html

Minimum rank of small of graphshttp://aimath.org/pastworkshops/catalog1.html


Developments

Leslie Hogben

3 mr problems

Simple trees

Loop trees

Ditrees

Computing mrditrees

Ditree proof


Symmetric mr

Related problems

Related problems

I Could consider asymmetric/diagonal free matricesdescribed by simple digraphs

I Could consider rectangular patterns (describingasymmetric matrices with diagonal constrained) -limited amount done

I Could consider other families of matrices such aspositive semidefinite - some of the symmetric/diagonalfree results have been extended to positive definite.

I Could consider matrices over other fields - many of thesymmetric/diagonal free results been extended to otherfields.


Developments

Leslie Hogben

3 mr problems

Simple trees

Loop trees

Ditrees

Computing mrditrees

Ditree proof


Symmetric mr

Related problems

Thank You!

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Minimum Rank Problems · Symmetric minimum rank problem Determine mr(G) for any simple graph G....

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